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We study meromorphic modular forms associated with positive definite binaryquadratic forms and their cycle integrals along closed geodesics in the modularcurve. We show that suitable linear combinations of these meromorphic modularforms have rational cycle integrals. Along the way, we evaluate the cycleintegrals of the Siegel theta function associated with an even lattice ofsignature $(1,2)$ in terms of Hecke's indefinite theta functions, which is ofindependent interest.

Cycle integrals of meromorphic modular forms and Siegel theta functions